Gary Coffman writes:

Therefore, the heat in the regulator
*is* directly proportional to the regulator temperature.

The system in view consists of the regulator and the air, not just the
regulator. If the air expands, this system can decrease in temperature
while gaining heat. And furthermore, the air flow can segregate the heat
and temperatures, so some parts of the system get colder and some hotter
(in temperature).